A Level Set Method in Shape and Topology Optimization for Variational Inequalities

Fulmański, Piotr; Laurain, Antoine; Scheid, Jean-Francois; Sokołowski, Jan

doi:10.2478/v10006-007-0034-z

Abstract

The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology variations in the form of small holes. The derivation of topological derivatives is performed within the framework proposed in (Sokołowski and Żochowski, 2003). Numerical results confirm that the method is efficient and gives better results compared with the classical shape optimization techniques.

References

Allaire G., De Gournay F., Jouve F. and Toader A.M. (2005): Structural optimization using topological and shape sensitivity via a level set method.Control and Cybernetics, Vol. 34, No. 1, pp. 59-80.
Search in Google Scholar
Amstutz S. and Andrä H. (2006): A new algorithm for topology optimization using a level-set method. — Journal of Computer Physics, Vol. 216, No. 2, pp. 573-588.10.1016/j.jcp.2005.12.015
Search in Google Scholar
Delfour M.C. and Zolesio J.-P. (2001): Shapes and Geometries. Philadelphia, PA: SIAM.
Search in Google Scholar
Henrot A. and Pierre M. (2005): Variation et optimisation de formes: Une analyse géométrique. Berlin: Springer.10.1007/3-540-37689-5
Search in Google Scholar
Jackowska L., Sokołowski J., Żochowski A. and Henrot A. (2002): On numerical solution of shape inverse problems. — Computational Optimization and Applications, Vol. 23, No. 2, pp. 231-255.10.1023/A:1020528902875
Search in Google Scholar
Jackowska A.L., Sokołowski J. and Żochowski A. (2003): Topological optimization and inverse problems.Computer Assisted Mechanics and Engineering Sciences, Vol. 10, No. 2, pp. 163-176.
Search in Google Scholar
Jarusek J., Krbec M., Rao M. and Sokołowski J. (2003): Conical differentiability for evolution variational inequalities.Journal of Differential Equations, Vol. 193, No. 1, pp. 131-146.10.1016/S0022-0396(03)00136-0
Search in Google Scholar
Laurain A. (2006): Singularly perturbed domains in shape optimization.—Ph.D. thesis, Université de Nancy.
Search in Google Scholar
Masmoudi M. (2002): The topological asymptotic, In: Computational Methods for Control Applications (R. Glowinski, H. Kawarada and J. Periaux, Eds.). GAKUTO Int. Ser. Math. Sci. Appl., Vol. 16, pp. 53-72
Search in Google Scholar
Maz'ya V., Nazarov S.A. and Plamenevskij B. (2000): Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vols. 1 and 2, Basel: Birkhäuser, p. 435.
Search in Google Scholar
Nazarov S.A. (1999): Asymptotic conditions at a point, self adjoint extensions of operators, and the method of matched asymptotic expansions.American Mathematical Society Translations, Vol. 198, No. 2, pp. 77-125.10.1090/trans2/193/05
Search in Google Scholar
Nazarov S.A. and Sokołowski J. (2003a): Self adjoint extensions of differential operators in application to shape optimization.Comptes Rendus Mécanique, Vol. 331, No. 10, pp. 667-672.10.1016/j.crme.2003.07.003
Search in Google Scholar
Nazarov S.A. and Sokołowski J. (2003b): Asymptotic analysis of shape functionals.Journal de Mathématiques pures et appliquées, Vol. 82, No. 2, pp. 125-196.10.1016/S0021-7824(03)00004-7
Search in Google Scholar
Nazarov S.A. and Sokołowski J. (2004a): Self adjoint extensions for elasticity system in application to shape optimization.Bulletin of the Polish Academy of Sciences, Mathematics, Vol. 52, No. 3, pp. 237-248.10.4064/ba52-3-4
Search in Google Scholar
Nazarov S.A. and Sokołowski J. (2004b): The topological derivative of the Dirichlet integral due to formation of a thin ligament.Siberian Mathematical Journal, Vol. 45, No. 2, pp. 341-355.10.1023/B:SIMJ.0000021290.61807.4b
Search in Google Scholar
Nazarov S.A., Slutskij A.S. and Sokołowski J. (2005): Topological derivative of the energy functional due to formation of a thin ligament on a spatial body.Folia Mathematicae, Acta Universitatis Lodziensis, Vol. 12, pp. 39-72.
Search in Google Scholar
Osher S. and Fedkiw R. (2004): Level Set Methods and Dynamic Implicit Surfaces. New York: Springer.10.1007/b98879
Search in Google Scholar
Osher S. and Sethian J. (1988): Fronts propagating with curvature-dependant speed: Algorithms based on Hamilton-Jacobi formulation.Journal of Computational Physics, Vol. 79, No. 1, pp. 12-49.10.1016/0021-9991(88)90002-2
Search in Google Scholar
Peng D., Merriman B., Osher S., Zhao S. and Kang M. (1999): A PDE-based fast local level set method.Journal of Computational Physics, Vol. 155, No. 2, pp. 410-438.10.1006/jcph.1999.6345
Search in Google Scholar
Rao M. and Sokołowski J. (2000): Tangent sets in Banach spaces and applications to variational inequalities. Les prépublications de l'Institut Élie Cartan, No. 42.
Search in Google Scholar
Sethian J. (1996): Level Set Methods. Cambridge: Cambridge University Press.
Search in Google Scholar
Sokołowski J. and Zolesio J.-P. (1992): Introduction to shape optimization. Series in Computationnal Mathematics, Berlin: Springer Verlag, Vol. 16.
Search in Google Scholar
Sokołowski J. and Żochowski A. (1999): On the topological derivative in shape optimization.SIAM Journal on Control and Optimization, Vol. 37, No. 4, pp. 1251-1272.10.1137/S0363012997323230
Search in Google Scholar
Sokołowski J. and Żochowski A. (2001): Topological derivatives of shape functionals for elasticity systems.Mechanics of Structures and Machines, Vol. 29, No. 3, pp. 333-351.10.1007/978-3-0348-8148-7_19
Search in Google Scholar
Sokołowski J. and Żochowski A. (2003): Optimality conditions for simultaneous topology and shape optimization.SIAM Journal on Control and Optimization, Vol. 42, No. 4, pp. 1198-1221.10.1137/S0363012901384430
Search in Google Scholar
Sokołowski J. and Żochowski A. (2005a): Topological derivatives for contact problems.Numerische Mathematik, Vol. 102, No. 1, pp. 145-179.10.1007/s00211-005-0635-0
Search in Google Scholar
Sokołowski J. and Żochowski A. (2005b): Topological derivatives for obstacle problems. Les prépublications de l'Institut Élie Cartan No. 12.
Search in Google Scholar
Watson G.N. (1944): Theory of Bessel Functions. Cambridge: The University Press.
Search in Google Scholar
Zhao H.K., Chan T., Merriman B. and Osher S. (1996): A variational level set approach to multi-phase motion.Journal of Computational Physics, Vol. 127, No. 1, pp. 179-195.10.1006/jcph.1996.0167
Search in Google Scholar

A Level Set Method in Shape and Topology Optimization for Variational Inequalities

Abstract

Paradigm

My account